We explore the following problem: given a collection of creases on a piece of
paper, each assigned a folding direction of mountain or valley, is there a flat
folding by a sequence of simple folds? There are several models of simple
folds; the simplest one-layer simple fold rotates a portion of paper
about a crease in the paper by ±180°. We first consider the
analogous
questions in one dimension lower—bending a segment into a flat
object—which
lead to interesting problems on strings. We develop efficient algorithms for
the recognition of simply foldable 1-D crease patterns, and reconstruction of a
sequence of simple folds. Indeed, we prove that a 1-D crease pattern is
flat-foldable by any means precisely if it is by a sequence of one-layer simple
folds.

Next we explore simple foldability in two dimensions, and find a surprising
contrast: “map” folding and variants are polynomial, but slight
generalizations are NP-complete. Specifically, we develop a linear-time
algorithm for deciding foldability of an orthogonal crease pattern on a
rectangular piece of paper, and prove that it is (weakly) NP-complete to decide
foldability of (1) an orthogonal crease pattern on a orthogonal piece of
paper,
(2) a crease pattern of axis-parallel and diagonal (45-degree) creases on a
square piece of paper, and (3) crease patterns without a mountain/valley
assignment.